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<Chapter Label = "Attributes and operations for semigroups">
  <Heading>
    Attributes and operations for semigroups
  </Heading>

  In this chapter we describe the methods that are available in &SEMIGROUPS; for
  determining the attributes of a semigroup, and the operations which can be
  applied to a semigroup.

  
  

  <Section>
    <Heading>
       Accessing the elements of a semigroup
     </Heading>
     <#Include Label = "AsListCanonical">
     <#Include Label = "PositionCanonical">
     <#Include Label = "Enumerate">
     <#Include Label = "IsEnumerated">
   </Section>

  
  

   <Section>
    <Heading>
       Cayley graphs
     </Heading>
     <#Include Label = "RightCayleyDigraph">
   </Section>

  
  

  <Section>

    <Heading>
      Random elements of a semigroup
    </Heading>

    <#Include Label = "Random">
  </Section>

  
  

  <Section>

    <Heading>
       Properties of elements in a semigroup
    </Heading>

    <#Include Label = "IndexPeriodOfSemigroupElement">
    <#Include Label = "SmallestIdempotentPower">
  </Section>

  
  

  <Section>

    <Heading>
       Operations for elements in a semigroup
    </Heading>
    <#Include Label = "OneInverseOfSemigroupElement">
  </Section>

  
  

  <Section>

    <Heading>
      Expressing semigroup elements as words in generators
    </Heading>

    It is possible to express an element of a semigroup as a word in the
    generators of that semigroup. This section describes how to accomplish this
    in &SEMIGROUPS;.<P/>

    <#Include Label = "EvaluateWord">
    <#Include Label = "Factorization">
    <#Include Label = "MinimalFactorization">
    <#Include Label = "NonTrivialFactorization">

  </Section>

  
  

  <Section>

    <Heading>
      Generating sets
    </Heading>

    <#Include Label = "Generators">
    <#Include Label = "SmallGeneratingSet">
    <#Include Label = "IrredundantGeneratingSubset">
    <#Include Label = "MinimalGeneratingSet">
    <#Include Label = "GeneratorsSmallest">
    <#Include Label = "IndecomposableElements">

  </Section>

  
  

  <Section>

    <Heading>
      Minimal ideals and multiplicative zeros
    </Heading>

    In this section we describe the attributes of a semigroup that can be found
    using the &SEMIGROUPS; package.
    <P/>

    <#Include Label = "MinimalIdeal">
    <#Include Label = "RepresentativeOfMinimalIdeal">
    <#Include Label = "MultiplicativeZero">
    <#Include Label = "UnderlyingSemigroupOfSemigroupWithAdjoinedZero">

  </Section>

  
  

  <Section>

    <Heading>
      Group of units and identity elements
    </Heading>

    <#Include Label = "GroupOfUnits">
    

  </Section>

  
  

  <Section>

    <Heading>
      Idempotents
    </Heading>

    <#Include Label = "Idempotents">
    <#Include Label = "NrIdempotents">
    <#Include Label = "IdempotentGeneratedSubsemigroup">

  </Section>

  
  

  <Section Label = "Computing maximal subsemigroups">

    <Heading>
      Maximal subsemigroups
    </Heading>

    The &SEMIGROUPS; package provides methods to calculate the maximal
    subsemigroups of a finite semigroup, subject to various conditions.  A
    <E>maximal subsemigroup</E> of a semigroup is a proper subsemigroup
    that is contained in no other proper subsemigroup of the semigroup.

    <P/>

    When computing the maximal subsemigroups of a regular Rees (0-)matrix
    semigroup over a group, additional functionality is available. As described
    in <Cite Key="Graham1968aa"/>, a maximal subsemigroup of a finite regular
    Rees (0-)matrix semigroup over a group is one of 6 possible types. Using the
    &SEMIGROUPS; package, it is possible to search for only those maximal
    subsemigroups of certain types.

    <P/>

    A maximal subsemigroup of such a Rees (0-)matrix semigroup <C>R</C> over a
    group <C>G</C> is either:

    <Enum>
      <Item><C>{0};</C></Item>
      <Item>formed by removing <C>0</C>;</Item>
      <Item>formed by removing a column (a non-zero &L;-class);</Item>
      <Item>formed by removing a row (a non-zero &R;-class);</Item>
      <Item>formed by removing a set of both rows and columns;</Item>
      <Item>isomorphic to a Rees (0-)matrix semigroup of the same dimensions over
        a maximal subgroup of <C>G</C> (in particular, the maximal subsemigroup
        intersects every &H;-class of <C>R</C>).</Item>
    </Enum>

    Note that if <C>R</C> is a Rees matrix semigroup then it has no
    maximal subsemigroups of types 1, 2, or 5. Only types 3, 4, and 6
    are relevant to a Rees matrix semigroup.

    <#Include Label = "MaximalSubsemigroups">
    <#Include Label = "NrMaximalSubsemigroups">
    <#Include Label = "IsMaximalSubsemigroup">

  </Section>

  
  

  <Section>

    <Heading>
      Attributes of transformations and transformation semigroups
    </Heading>

    <#Include Label = "ComponentRepsOfTransformationSemigroup">
    <#Include Label = "ComponentsOfTransformationSemigroup">
    <#Include Label = "CyclesOfTransformationSemigroup">
    <#Include Label = "DigraphOfAction">
    <#Include Label = "DigraphOfActionOnPoints">
    <#Include Label = "FixedPointsOfTransformationSemigroup">
    <#Include Label = "IsTransitive">
    <#Include Label = "SmallestElementSemigroup">
    <#Include Label = "CanonicalTransformation">
    <#Include Label = "IsConnectedTransformationSemigroup">

  </Section>

  
  

  <Section>

    <Heading>
      Attributes of partial perm semigroups
    </Heading>

    <#Include Label = "ComponentRepsOfPartialPermSemigroup">
    <#Include Label = "ComponentsOfPartialPermSemigroup">
    <#Include Label = "CyclesOfPartialPerm">
    <#Include Label = "CyclesOfPartialPermSemigroup">

  </Section>

  
  

  <Section>

    <Heading>
      Attributes of Rees (0-)matrix semigroups
    </Heading>

    <#Include Label = "RZMSDigraph">
    <#Include Label = "RZMSConnectedComponents">

  </Section>

  <Section>

    <Heading>
      Attributes of inverse semigroups
    </Heading>

    <#Include Label = "NaturalLeqInverseSemigroup">
    <#Include Label = "JoinIrreducibleDClasses">
    <#Include Label = "MajorantClosure">
    <#Include Label = "Minorants">
    <#Include Label = "PrimitiveIdempotents">
    <#Include Label = "RightCosetsOfInverseSemigroup">
    <#Include Label = "SameMinorantsSubgroup">
    <#Include Label = "SmallerDegreePartialPermRepresentation">
    <#Include Label = "VagnerPrestonRepresentation">
    <#Include Label = "CharacterTableOfInverseSemigroup">
    <#Include Label = "EUnitaryInverseCover">

  </Section>

  
  

  <Section>

    <Heading>
      Nambooripad partial order
    </Heading>

    <#Include Label = "NambooripadLeqRegularSemigroup">
    <#Include Label = "NambooripadPartialOrder">

  </Section>

  
  

</Chapter>

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